In many cases the surface velocity is assumed to be only tangential to the squirmer surface and
time-independent.
This describes in a good approximation the hydrodynamic flows close to spherical artificial microswimmers.
Examples are active colloids [1]}, [2]} and droplets [3]}, [4]}, [5]}
which move themselves forward realizing static tangential stresses close to their surface,
for example induced by self-phoretic mechanisms or self-induced Marangoni flows, respectively.
Other examples are biological microswimmers such as
Paramecia [6]} or Volvox [7]}
which propel themselves by synchronously beating thousands of tiny cilia attached to their surface.
Averaged over a beating cycle, they induce
strong tangential flows to move forward.
Setting \(v_r(\theta ,t)=0\) , the velocity field \(\mathbf {v}(\mathbf {r})\) 
around a squirmer of radius \(R\)  with orientation \(\mathbf {e}\) 
and
located at position \(\mathbf {r}_0\)  can be calculated to [8]}
\(\begin{split}& \mathbf {v}(\mathbf {r}) = -\frac{1}{3} \frac{R^3}{r^3}B_1 \mathbf {e}+ \frac{R^3}{r^3}B_1(\mathbf {e} \cdot \hat{\mathbf {r}})\hat{\mathbf {r}} \\& + \sum _{n=2}^{\infty } \left( \frac{R^{n+2}}{r^{n+2}}- \frac{R^{n}}{r^{n}}\right)B_n P_n (\mathbf {e} \cdot \hat{\mathbf {r}})\hat{\mathbf {r}} \\& + \sum _{n=2}^{\infty } \left( \frac{n}{2} \frac{R^{n+2}}{r^{n+2}} - \left(\frac{n}{2} -1 \right)\frac{R^{n}}{r^{n}}\right)B_nW_n(\mathbf {e} \cdot \hat{\mathbf {r}}) \left[(\mathbf {e} \cdot \hat{\mathbf {r}})\hat{\mathbf {r}}-\mathbf {e}\right].\end{split}\) 
